The cross-section of a tunnel, perpendicular to its length is a trapezium ABCD in which AB = 8 m, DC = 6 m and AL = BM. The height of the tunnel is 2.4 m and its length is 40 m. Find :
(i) the cost of paving the floor of the tunnel at ₹ 16 per m².
(ii) the cost of painting the internal surface of the tunnel, excluding the floor at the rate of ₹ 5 per m².

Question

The cross-section of a tunnel, perpendicular to its length is a trapezium ABCD in which AB = 8 m, DC = 6 m and AL = BM. The height of the tunnel is 2.4 m and its length is 40 m. Find :
(i) the cost of paving the floor of the tunnel at ₹ 16 per m².
(ii) the cost of painting the internal surface of the tunnel, excluding the floor at the rate of ₹ 5 per m².

KnowledgeBoat · Accepted Answer

Given, AB = 8 m DC = 6 m Height = 2.4 m Length of the tunnel = 40 m AL = BM From figure, LM = DC = 6 m (since perpendiculars drop from D and C). From figure, ⇒ AB = AL + LM + MB ⇒ 8 = AL + 6 + BM Since, AL = BM ∴ 8 = 2AL + 6 ⇒ 2AL = 8 - 6 ⇒ 2AL = 2 ⇒ AL = = 1 m ∴ AL = BM = 1 m. Calculating the lengths of the sloping sides AD and BC, In right triangle ALD, AL = 1 m DL = 2.4 m Using pythagoras theorem for the triangle ALD, ⇒ Hypotenuse2 = Base2 + Height2 ⇒ AD2 = AL2 + DL2 ⇒ AD2 = 12 + (2.4)2 ⇒ AD2 = 1 + 5.76 ⇒ AD2 = 6.76 ⇒ AD = ⇒ AD = 2.6 m. ∴ AD = BC = 2.6 m. (i) Cost of paving the floor: Area of tunnel = AB × length = 8 × 40 = 320 m2. Total cost = Area × cost per m2 = 320 × 16 = ₹ 5,120. Hence, cost of paving the floor = ₹ 5,120. (ii) Cost of painting internal surface (excluding floor) Area to be painted are : Roof DC and Two sloping sides AD and BC. Calculating the area of DC, Area = length × DC = 40 × 6 = 240 m2. Calculating the area of slope AD, Area = length × AD = 40 × 2.6 = 104 m2. Calculating the area of slope BC, Area = Length × BC = 40 × 2.6 = 104 m2. Total area = 240 + 104 + 104 = 448 m2. Cost rate = ₹ 5 per m2 Calculating the cost of painting the internal surface of the tunnel, Total cost = Total area × Cost = 448 × 5 = ₹ 2,240. Hence, cost = ₹ 2,240.

Mathematics

Mensuration

Related Questions

The area of cross-section of a rectangular pipe is 5.4 cm² and water is pumped out of it at the rate of 27 kmph. Find, in litres, the volume of water which flows out of the pipe in 1 minute.

The adjoining figure shows a solid of uniform cross section. Find the volume of the solid. It is being given that all the measurements are in cm and each angle in the figure is a right angle.

If the length of each side of a cube is reduced by 25%, then the ratio of the volumes of the original and the new cube is :
64 : 1
4 : 3
64 : 27
32 : 9

If the length of each side of a cube is reduced by 50%, then the ratio of the total surface area of the original and the new cube is :
2 : 1
4 : 1
8 : 1
8 : 3